Could you be a little more specific? What version of Android did you update your Galaxy 10.1--was it 4.2? After the update, is it only YouTube that isn't working, or are you having other connectivity issues as well?
General things to try:
1. Reboot your router by unplugging it for 10 seconds, then plugging it back in again. Wait about 30 seconds before the internet becomes available again.
2. Go to Settings/Wi-fi, longpress on your wi-fi hotspot, and forget the settings. Then login again.
3. Compare with other streaming video services like Google Movies or Vimeo--are videos working on those sites or apps? If not, then it's probably more of a global problem.